Piezoelectric MEMS microphone

ABSTRACT

A microphone including a casing having a front wall, a back wall, and a side wall joining the front wall to the back wall, a transducer mounted to the front wall, the transducer including a substrate and a transducing element, the transducing element having a transducer acoustic compliance dependent on the transducing element dimensions, a back cavity cooperatively defined between the back wall, the side wall, and the transducer, the back cavity having a back cavity acoustic compliance. The transducing element is dimensioned such that the transducing element length matches a predetermined resonant frequency and the transducing element width, thickness, and elasticity produces a transducer acoustic compliance within a given range of the back cavity acoustic compliance.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.14/208,827 filed 13 Mar. 2014, which claims the benefit of U.S.Provisional Application No. 61/779,892 filed 13 Mar. 2013, and which isa continuation-in-part of U.S. application Ser. No. 13/963,661 filed 9Aug. 2013, now U.S. Pat. No. 8,896,184, which is a continuation of U.S.application Ser. No. 12/495,195 filed 30 Jun. 2009, now U.S. Pat. No.8,531,088, which claims the benefit of U.S. Provisional Application No.61/076,928 filed 30 Jun. 2008. The entire contents of each of theseprior applications are hereby incorporated by reference.

TECHNICAL FIELD

This invention relates generally to piezoelectric microphones and, inparticular, to piezoelectric MEMS microphones and design techniques forconstructing such microphones to meet the requirements of a particularend use application.

BACKGROUND

The rise of microelectromechanical systems (MEMS) technology has enabledthe development of acoustic transducers such as microphones usingsilicon-wafer deposition techniques. Microphones fabricated this way arecommonly referred to as MEMS microphones and can be made in variousforms such as capacitive microphones or piezoelectric microphones usinga material such as PZT, ZnO, PVDF, PMN-PT, or AlN. MEMS capacitivemicrophones and electret condenser microphones (ECMs) are used inconsumer electronics and have an advantage over typical piezoelectricMEMS microphones in that they have greater sensitivity and lower noisefloors. However, each of these more ubiquitous technologies has its owndisadvantages. For standard ECMs, they typically cannot be mounted to aprinted circuit board using the typical lead-free solder processingcommonly used on all other microchips attached to the board. MEMScapacitive microphones, which are often used in cell phones, arerelatively expensive due at least in part to the use of anapplication-specific integrated circuit (ASIC) that provides readoutcircuitry for the microphone. MEMS capacitive microphones also have asmaller dynamic range than typical piezoelectric MEMS microphones.

The noise floors of various known piezoelectric and capacitive MEMSmicrophones are shown in FIG. 1. As indicated by the two encircledgroups of microphones, capacitive MEMS microphones (the lower group)generally have a noise floor that is about 20 dB lower than similarlysized piezoelectric MEMS microphones.

Known piezoelectric MEMS microphones have been made either ascantilevered beams or as a diaphragm, and these microphones include bothelectrodes and the piezoelectric material along with a structuralmaterial such as Parylene or silicon that is used as a diaphragm or beamsubstrate material. An advantage of Parylene for cantilever designs isthat it is can be used to increase the thickness of the beam which bothincreases the bandwidth of the beam (for a fixed length) and increasesthe distance from the neutral axis of the piezoelectric material, whichseemingly increases sensitivity. For example, beam substrates of about20 μm are known, see Ledermann [15]. For piezoelectric MEMS microphonesthat utilize a Parylene diaphragm, thinner layers have been used. See,for example, U.S. Pat. No. 6,857,501 and Niu [10]. Note that the variousreferences made herein to other authors are references to literature andjournal articles identified at the end of this description and areprovided only for non-essential subject matter in support of or asbackground for some of the teachings herein. Each of the referencedworks are hereby incorporated by reference.

Conventional microphones additionally suffer from high noise floors dueto the limited back cavity volume of the increasingly smallermicrophones. As taught by the prior art, noise levels are affected bythe back cavity stiffness of the microphone. More specifically, noiselevels are conventionally reduced by maximizing the back cavity volumeto near-infinite volumes, thereby minimizing the back cavity stiffness.While back cavity volume maximization to near-infinite volumes isfeasible in larger microphone systems, this type of maximization cannotbe realized in smaller microphone systems, particularly in microphonessmaller than 50 mm³, where any increase in volume comes at a premium.Without the ability to minimize back cavity stiffness, conventionalmicrophone designs either ignore the effects of the back cavity volumeon microphone performance, or account for the back cavity stiffness as aparasitic stiffness.

Thus, there is a need in the piezoelectric MEMS acoustic transducerfield to create a new and useful acoustic transducer with low frequencysensitivity despite residual stresses. Furthermore, there is a need inthe acoustic transducer field for a new and useful acoustic transducerdesign that optimizes the transducer output for a given back cavityvolume.

SUMMARY OF THE INVENTION

In accordance with one aspect of the invention, there is provided apiezoelectric MEMS microphone, comprising a substrate and a multi-layeracoustic sensor having at least three layers that includes a firstelectrode layer, an intermediate layer of piezoelectric materialdeposited over the first electrode layer, and a second electrode layerdeposited over the piezoelectric material. The sensor is dimensionedsuch that the ratio of output energy to sensor area for the multi-layersensor is at least 10% of the maximum ratio obtainable for a given inputpressure, bandwidth, and piezoelectric material.

In accordance with another aspect of the invention, there is provided apiezoelectric MEMS microphone, comprising a substrate and a multi-layeracoustic sensor having at least three layers that includes a firstelectrode layer, an intermediate layer of piezoelectric materialdeposited over the first electrode layer, and a second electrode layerdeposited over the piezoelectric material. The sensor is dimensionedsuch that an optimization parameter calculated according to the equation

${{Optimization}\mspace{14mu}{Parameter}} = {\frac{V_{out}^{2}C}{P^{2}A\;{\tan(\delta)}} \cdot f_{res}^{2}}$is at least 10% of the maximum obtainable optimization parameter for thesensor, where V_(out) is output voltage of the sensor, C is thecapacitance of the sensor, P is the input pressure, A is the sensorarea, tan(∂) is dielectric loss angle of the sensor at the sensor'sfirst resonant frequency, and f_(res) is the first resonant frequency.

In accordance with another aspect of the invention, there is provided apiezoelectric MEMS microphone, comprising a silicon substrate and aplurality of beams each supported at one end by the substrate such thateach beam is cantilevered and extends between a fixed end and a freeend. Each beam comprises a deposited layer of electrode material and adeposited layer of piezoelectric material overlying the electrodematerial. At least some of the beams are stacked such that the stackedbeams include alternative layers of deposited electrode material anddeposited piezoelectric material with no additional layers therebetween.The beams are preferably arranged such that an edge of each beam issubstantially parallel to an edge of the adjacent beam (e.g., within atolerance, such as within 2 microns, 5 microns, 20% of the beamthickness, 10% of the beam thickness, 5% of the beam thickness, etc.).

In accordance with yet another aspect of the invention, there isprovided a piezoelectric MEMS microphone, comprising a substrate and astress-relieved diaphragm suspended above the substrate. The diaphragmcomprises a multi-layer acoustic sensor having at least three layersincluding a first electrode layer, an intermediate layer ofpiezoelectric material deposited over the first electrode layer, and asecond electrode layer deposited over the piezoelectric material. Thestress relieved diaphragm can be obtained in any suitable manner, suchas for example, by detaching it from the substrate about substantiallyall of its periphery and allowing it to expand or contract as needed torelieve residual stress. The diaphragm can then be reattached to thesubstrate about its periphery by any suitable technique.

BRIEF DESCRIPTION OF THE FIGURES

One or more preferred exemplary embodiments of the invention willhereinafter be described in conjunction with the appended drawings,wherein like designations denote like elements, and wherein:

FIG. 1 is a plot of noise levels versus sensor area for various knownMEMS microphones;

FIG. 2 is a plot depicting the influence of diaphragm residual stress onoutput energy of a piezoelectric MEMS microphone;

FIG. 3a is a top view of a beam cantilever piezoelectric MEMS microphonesensor constructed in accordance with one aspect of the invention;

FIG. 3b depicts a cross-sectional view of two pairs of facing beams fromthe microphone sensor of FIG. 3 a;

FIG. 3c shows alternating beam layers and their dimensions for use inmodeling the behavior of the stacked beams shown in FIG. 3 b;

FIG. 4 depicts a schematic of the microphone of FIG. 3a connected to anamplifying circuit, showing impedance modeling for the circuit;

FIG. 5 is a plot of typical noise curves for a piezoelectric acousticsensor;

FIG. 6 depicts the impact of beam taper on the output energy of thesensor of FIG. 3 a;

FIG. 7 depicts the impact of including one or more Paralyene layersshowing the impact of layer thickness on the output energy of apiezoelectric MEMS microphone sensor;

FIG. 8 is a plot showing how different electrode materials affect theenergy output of a piezoelectric MEMS microphone sensor;

FIGS. 9a-9d depict the processing steps used to make the sensor of FIG.3 b;

FIG. 10 is a microscope picture of a fabricated sensor of FIG. 3 a;

FIG. 11 is a picture of a piezoelectric MEMS microphone using the sensorof FIG. 3 a;

FIG. 12 shows plots of the frequency response of the microphone of FIG.11;

FIG. 13 is a plot of the beam deflection profile of the microphone ofFIG. 11;

FIG. 14 is a plot of the measured and predicted sensitivities and noisefloors for the microphone of FIG. 11;

FIG. 15 is a plot of the normalized output energy as a function ofelectrode length for a cantilever beam of the type shown in FIG. 3 a;

FIG. 16 is a plot showing the degradation of the d₃₃ coefficient fromthe piezoelectric coupling coefficient matrix for the AlN piezoelectricmaterial;

FIG. 17 is a plot showing the degradation of the dielectric loss angletan(∂);

FIG. 18 is a plot showing Mo resistivity as a function of the electrodelayer thickness;

FIG. 19 is a plot of the relationship between piezoelectric layerthickness and the d₃₁ coefficient from the piezoelectric couplingcoefficient matrix for the AlN piezoelectric material;

FIG. 20 is a plot showing the dielectric loss angle as a function of AlNlayer thickness;

FIG. 21 depicts the calculated optimization parameter as a function ofMo bottom electrode layer thickness for a single (non-stacked)cantilever beam;

FIG. 22 depicts the calculated optimization parameter as a function ofAlN intermediate layer thickness for a single (non-stacked) cantileverbeam;

FIG. 23 depicts the calculated optimization parameter as a function ofMo upper electrode layer thickness for a single (non-stacked) cantileverbeam;

FIG. 24 depicts the calculated optimization parameter as a function ofMo bottom and top electrode layer thickness for a five layer (stacked)cantilever beam;

FIG. 25 depicts the calculated optimization parameter as a function ofAlN intermediate layer thickness for a five layer (stacked) cantileverbeam;

FIG. 26 depicts the calculated optimization parameter as a function ofMo middle electrode layer thickness for a five layer (stacked)cantilever beam;

FIG. 27a is a top view of a diaphragm piezoelectric MEMS microphonesensor constructed in accordance with one aspect of the invention;

FIG. 27b is a partial cross-sectional view taken along the B-B line ofFIG. 27 a;

FIG. 28 is a plot of expected noise floors for piezoelectric MEMSmicrophones constructed in accordance with the invention, showing howthey compare to known piezoelectric and capacitive MEMS microphones;

FIG. 29 is a schematic representation of a microphone with a transducerand casing defining a back cavity;

FIG. 30 is a perspective view of a rectangular transducer beam withelectrodes extending over half of the beam area;

FIG. 31 is a top-down view of a circular diaphragm with electrodesextending over the central quarter and peripheral quarter of thediaphragm area; and

FIG. 32 is a perspective view of a triangular transducer beam withelectrodes extending over two-thirds of the diaphragm area.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The description that follows is directed to various embodiments of apiezoelectric MEMS microphone that meets an optimization criteria whichcan be determined in one or more of the different manners describedbelow.

Typical piezoelectric MEMS microphones are designed so as to optimizesensitivity of the microphone, and this is at least partiallyresponsible for the increased noise floors noted above for thesedevices. As described below, by optimizing the ratio of output energy tosensor area for a given input pressure, bandwidth, and piezoelectricmaterial, a piezoelectric MEMS microphone can be constructed that hassufficient sensitivity for typical applications along with a noise floorsimilar to that of capacitive MEMS microphones. This approach is validfor films of good quality. However, as the film is reduced in thickness,the film quality will degrade. This factor can be accounted for with analternative approach described herein which utilizes a calculatedoptimization parameter that is still a ratio of sensor energy to area,but also includes the pressure, natural frequency (which limits thebandwidth), and the loss angle of the device. By adding these parametersinto the calculated ratio, this alternative approach accounts for theeffects of these parameters, rather than considering them as constants.Thus, as will be appreciated by those skilled in the art, the followingembodiments are described in conjunction with two different usableapproaches for determining an optimal or near-optimal sensor design—1) astraight calculation of output energy to sensor area ratio for a given(constant) input pressure, bandwidth and piezoelectric material and 2) acalculation of an optimization parameter that accounts for the pressure,natural frequency (which limits the bandwidth), and the loss angle ofthe device. This optimization parameter can be determined using theequation:

${{Optimization}\mspace{14mu}{Parameter}} = {\frac{V_{out}^{2}C}{P^{2}A\;{\tan(\delta)}} \cdot f_{res}^{2}}$where V_(out) is the output voltage, C is the device capacitance, P isthe input pressure, A is the sensor area, tan(∂) is the dielectric lossangle or dissipation factor of the microphone at the first resonantfrequency, and f_(res) is the first resonant frequency of the device.The use of this optimization parameter and the material characteristicsand device geometry used in calculating this parameter will be describedfarther below. When the optimal film thicknesses are compared to thefilm properties plotted in FIGS. 18, 19, and 20 that are describedfurther below, it is clear that most optimal film thicknesses havevalues such that they have nearly the properties of a thick film. Forthese films, optimization of the calculated energy to sensor area alonemay be suitable without accounting for changes in electrode and/orpiezoelectric properties. However, making the films substantiallythinner than the optimal thicknesses can cause a large relative changein material properties, in which case use of the optimization parametermay be more suitable in determining sensor optimization.

There are several aspects of more conventional piezoelectric MEMSmicrophone designs that have typically kept these prior designs fromachieving optimization. A first is the use of sensor structures, such asdiaphragms, that have their stiffness dominated by tension. Forpiezoelectric MEMS microphone fabricated on a silicon wafer substrate,this tension is the result of residual stress left on each layer afterdeposition. This effect leads to a reduction in normalized outputenergy, one of the elements of the optimization parameter, as can beseen in FIG. 2. This figure shows how residual stress reduces thenormalized output energy of a diaphragm with two 1 μm aluminum nitride(AlN) layers and three 100 nm molybdenum (Mo) layers with 20 kHzresonant frequency. Stresses as low as 1 MPa, a stress level difficultto achieve, reduces the normalized output energy of this diaphragm by20%, reducing the optimization parameter by 20% as well.

A second problem with prior designs that have kept them from approachingoptimization is that these designs have not utilized optimal ornear-optimal device geometries. Thus, for example, with cantilevereddesigns for which device residual stress is not a significant factor(due to the device being released from its substrate when forming thecantilever), the combination of layer thickness, layer order, beamshape, and even beam spacing to adjacent beams, make up the overalldevice geometry for which optimization is desired.

Furthermore, the inventors have discovered that by designing thepackaged microphone to have a transducer acoustic compliance (e.g., MEMSmicrophone acoustic compliance) based on the back cavity acousticcompliance, the output energy of the microphone resulting from a fixedacoustic pressure can be controlled, thereby controlling microphonenoise. In particular, the inventors have discovered that matching themicrophone acoustic compliance with the back cavity acoustic compliancemaximizes the output energy of the microphone, thereby reducingmicrophone noise. The inventors have further discovered that the outputenergy for a microphone with a fixed back cavity volume does notincrease with an increase in the microphone acoustic compliance beyondthe back cavity acoustic compliance. While the prior art teaches thatnoise is minimized by making the back cavity volume as large aspossible, thereby making the back cavity compliance much larger than themicrophone acoustic compliance, the inventors have discovered that whendesigning with a fixed back cavity volume, noise is minimized byequating the microphone acoustic compliance with that of the backcavity.

Designers of conventional systems would not have been able to identifythis ideal microphone compliance for a given back cavity volume becauseconventional designers optimize the microphones for sensitivity (insteadof output energy or the product of microphone capacitance andsensitivity squared). Because microphone designers typically optimizemicrophones for sensitivity, they would not notice the relationshipbetween output energy and acoustic compliance. Designers of conventionalsystems typically build systems in which the stiffness of the backcavity has a relatively small effect on the sensitivity. In contrast,the inventors have found that noise is minimized when the transducer andback cavity are sized such that transducer sensitivity is approximatelyhalved relative to the transducer sensitivity of a system with aninfinite (or approximated infinite) back cavity. Because a microphonehaving these disclosed compliance ratios can have a higher or lowersensitivity than that of a microphone with a different ratio, designersof conventional systems are likely to bypass this microphone design.This microphone design is not obvious even to a designer optimizing foroutput energy, since conventional systems so heavily teach away fromlimiting the back cavity volume—based on the prior art, a designer wouldassume that the output would be maximized by designing the microphonesuch that the back cavity stiffness has a negligible affect on thesensitivity.

In accordance with the disclosed embodiments, these problems can beaddressed in one or more ways. For cantilevers where device residualstress is not problematic, this can be done by utilizing a microphonedesign that achieves at least 10% of the maximum ratio obtainable ofoutput energy to sensor area for a given input pressure, bandwidth, andpiezoelectric material. As used herein, the “maximum ratio obtainable”for a given sensor design can be determined using an output energycalculation along with sensor area, or can be determined using theoptimization parameter equation given above along with available (albeitsometimes varying) values and equations for the various parameters usedin the optimization equation. In this latter approach, suitable sensordesigns can be obtained for which the calculated optimization parameteris at least 10% of the maximum obtainable optimization parameter for thesensor. Other ways of determining the maximum ratio obtainable arepossible, such as by repeated experimental determination or by usingother optimization equations or techniques that are either now known orlater developed. To achieve the desired level of 10% or more of themaximum optimization attainable, it has been determined through modelingand subsequent prototype testing that it is beneficial to make thesensor nearly as thin as possible and employing it in a topology inwhich multiple beams are either stacked to increase output or are builtas individual beams with a thin (˜1 μm) Parylene layer located centrallybetween the electrode and crystalline layers. In either approach, aplurality of beams can be produced that are then wired in a combinationof series and parallel connections to obtain a desired combination ofdevice capacitance and sensitivity for any given application. Fordiaphragms, an improved piezoelectric MEMS microphone can be built usinga stress-relieved diaphragm, in which the piezoelectric sensor is madeby deposition on a silicon-based substrate, then released from thesubstrate to permit expansion or contraction of the released membrane torelieve any residual stress, and then re-attached in any suitablemanner. This technique could work for a diaphragm with any combinationof clamped, pinned, or free perimeter conditions. Use of theoptimization calculation above can also be used in making the diaphragmpiezoelectric MEMS microphone to provide enhanced microphone sensitivityand noise performance. These cantilever and diaphragm designs provideuseful operation of the device for many applications, and designs forwhich the calculated optimization parameter is above 10% of optimal canprovide enhanced operation that provides good sensitivity with noisefloors per unit area that are on par with or exceeding that ofcapacitive MEMS microphones. Furthermore, the ratio of the transducercompliance to the back cavity compliance can be controlled to optimizeoutput from either cantilevered or diaphragm microphones. A transducerof the preferred embodiment can include a substrate and a transducingelement. The transducer preferably has a transducer compliance based onthe dimensions of the transducing element. For example, the transducercompliance for a beam transducer is based on the width of the beams, thelength of the beams, the modulus of elasticity of the beams, thePoisson's ratio of the beam, and the beam thickness. Alternatively, thetransducer compliance can be based on the surface area (e.g., asdetermined by the width and length, diameter, or any other suitabledimension), thickness, modulus of elasticity, and Poisson's ratio forany transducing element.

The acoustic compliance can be approximated by the integral oftransducer displacement (per unit pressure) over the transducer area.Alternatively, the acoustic compliance can be approximated by theaverage acoustic compliance over the transducer area, multiplied by thetransducer area. Alternatively, the acoustic compliance can beapproximated by the back cavity compliance at which the output energy isapproximately 25% (alternatively, larger or smaller) of that which isachieved with an infinite (or near infinite) back cavity volume.However, the acoustic compliance can be otherwise defined. The acousticcompliance can alternatively be empirically determined. In one example,the acoustic compliance can be measured by applying a given amount ofpressure (e.g., 1 Pa) to the transducer (e.g., MEMS chip) with a verylarge back cavity volume (e.g., open to the ambient environment) at agiven frequency (e.g., 1 kHz), and measuring the displacement on a grid(e.g., 100×100 point grid) over the transducer with a vibrometer (e.g.,laser vibrometer). The displacement measurements can be averaged andmultiplied by the transducer or chip area to obtain the acousticcompliance.

For example, the acoustic compliance of a substantially rectangularpiezoelectric transducer beam can be approximately calculated asfollows:

${Compliance} = \frac{3{bL}^{5}}{5\;\frac{E}{1 - v^{2}}h^{3}}$wherein b is the beam width, L is the beam length, E is the modulus ofelasticity of the beam, ν is the Poisson's ratio of the beam, and h isthe beam thickness. The acoustic compliance of the transducer ispreferably the sum of the acoustic compliances for each beam within thetransducer. Similarly, the acoustic compliance for a substantiallytriangular transducer beam or a diaphragm transducer can be approximatedby analogous equations. The acoustic compliance of the transducer ispreferably the sum of the acoustic compliances for each beam within thetransducer, but can alternatively be otherwise determined orapproximated. The acoustic compliance for other suitable transducingelements can be derived in a similar manner to those described above.

As shown in FIG. 29, the packaged microphone 10 includes a MEMSmicrophone chip (microphone) 100 having a first acoustic compliance anda casing 200, wherein the microphone 100 and casing 200 cooperativelydefine a back cavity 300 with a second acoustic compliance. The packagedmicrophone components are preferably dimensioned to achieve apredetermined ratio between the first and second acoustic compliances.The microphone 100 is preferably a piezoelectric transducer, but mayalternatively be a capacitive transducer, an optical transducer (e.g.optical acoustic sensor), or any other suitable transducer that suffersfrom stressed cantilevers. The preferred microphone 100 is preferably anacoustic transducer, more preferably an acoustic sensor, but mayalternatively be driven with a voltage or current and used as a speaker.The preferred microphone 100 is preferably incorporated into consumerelectronics, such as mobile phones, but can be used in medicalapplications (e.g. hearing aids), photoacoustic detection, ultrasonicapplications, or any other transducer-based application as a sensor orspeaker.

The transducer dimensions and materials are preferably selected to meeta predetermined ratio between the transducer compliance and the acousticcompliance for a given back cavity volume. The transducer compliance ispreferably 100% (e.g. 1:1) of the back cavity acoustic compliance, butcan alternatively be between 17% and 581% of the back cavity acousticcompliance (e.g., to achieve a noise floor within 3 dB of the minimum),between 24% and 410% of the back cavity acoustic compliance (e.g., toachieve a noise floor within 2 dB of the minimum), between 38% and 266%of the back cavity acoustic compliance (e.g., to achieve a noise floorwithin 1 dB of the minimum), or any other suitable ratio. The transducerdimensions can additionally be selected to match a desired resonantfrequency. For example, for a beam transducer, the beam length can beselected based on the resonant frequency and the width can be selectedbased on the beam length and the desired ratio.

The microphone 100 can include a substrate 160. The substrate 160 of themicrophone 100 functions to support the microphone 100 during themanufacturing process, as well as to support the cantilevered beam 120of the microphone 100 during operation. In one variation of themicrophone 100, the substrate 160 is composed at least in part ofsilicon or any suitable silicon-based compound or derivative (e.g. Siwafer, SOI, polySi on SiO2/Si, etc.) Alternately, the substrate 160 canbe composed at least in part of fiberglass, glass, or any suitablecombination of materials. In another variation of the microphone 100,the substrate 160 is substantially removed from the active area of thetransducing element during the manufacturing process such that thetransducing element has maximized travel and response.

The transducing element is preferably made from alternatingpiezoelectric and electrode layers 142, but can alternatively and/oradditionally be made from capacitive material, ferroelectric material,inductive material, metal ribbon, carbon, fiber optics, materials usedin MEMS processes, or any other suitable material. The piezoelectriclayers 144 can function to convert applied pressures to voltages, andthe electrode layers 142 can function to transmit the generated voltagesto an amplifier such as a JFET, a charge amplifier, or an integratedcircuit. The piezoelectric layers 144 preferably include aluminumnitride (AlN) due to its CMOS compatibility, but can alternativelyinclude lead zirconate titanate (PZT), zinc oxide (ZnO), polyvinylidenefluoride (PVDF), lead magnesium niobate-lead titanate (PMN-PT), or anyother suitable piezoelectric material. The electrode layers 142preferably include molybdenum (Mo), titanium (Ti), aluminum (Al), orplatinum (Pt), but can alternately include any other suitable electrodematerial.

The transducing element preferably includes two piezoelectric layers 144interspersed between three electrode layers 142. However, thetransducing element can include three piezoelectric layers 144interspersed between the three electrode layers 142, include only threetotal layers (a first electrode layer 142, a first piezoelectric layer144, and a top electrode layer 142), or any number of layers in anysuitable permutation of electrode layers 142 and piezoelectric layers144. Preferably, the transducing element incorporates at least onepiezoelectric layer 144 and one electrode layer 142. Additionally,although each electrode layer 142 preferably defines only oneindependent electrode per electrode layer 142, the electrode layers 142can be patterned to define multiple independent electrodes per electrodelayer 142. The electrode layers 142 are preferably coupled together bymetal traces in series, but can be coupled in parallel or both in seriesand in parallel.

The electrode layers 142 preferably cover a portion of the transducingelement 120 such that they maximize the output energy of the microphonedue to a given input acoustic pressure, but can alternatively extendover the entirety of the transducing element broad face. The electrodelayers 142 preferably extend along a transverse plane extending throughthe transducing element 120, wherein the transverse plane is preferablysubstantially parallel (e.g. within a threshold percentage or distanceof being parallel, such as within a tolerance range of 5 microns) to abroad face of the transducing element (e.g. along the surface area ofthe transducing element 120). The electrode layers can extend along asurface of the transducing element 120, extend within the thickness ofthe transducing element 120 substantially parallel the broad face of thetransducing element, or extend along any other suitable portion of thetransducing element. The electrode layers 142 preferably extend from theportion of the transducing element 120 proximal the substrate, such asfrom the beam base or the diaphragm perimeter, but can alternativelycover an intermediate portion of the transducing element 120, extendfrom the portion of the transducing element 120 distal the substrate(e.g. the beam tip or diaphragm center), or cover any other suitableportion of the transducing element 120. The optimal electrode coveragecan be determined by measuring the deflection of a MEMS chip with auniform acoustic pressure and optimizing the electrode shape based onthe output voltage and transducer capacitance (or on the measureddeflection profile). The electrode(s) preferably cover the portions ofthe beam, plate, or diaphragm with the highest stress, but canadditionally or alternatively cover the portions of the transducingelement with the lowest stress or any other suitable stress level. Asthe electrode size is increased to cover areas with lower stress, thecapacitance will increase but the sensitivity will decrease. Theelectrode is preferably arranged such that the product of capacitanceand sensitivity squared (the output energy) is maximized, but can beotherwise arranged. However, the electrode placement and coverage overthe transducing element can be otherwise determined.

For example, when the transducing element 120 includes a triangularbeam, the electrode layers preferably extend over two-thirds of the beamarea from the beam base 134, as shown in FIG. 32, but can alternativelyextend over more or less than two-thirds of the beam (e.g., less than100%, less than 80%, less than 70%, less than 30%, etc.), and can extendover any suitable portion of the beam. Alternatively, the electrodelayers can extend over two-thirds of the beam length from the beam base134. When the transducing element 120 includes a rectangular beam, theelectrode layers preferably extend over half of the beam broad face fromthe beam base 134 (as shown in FIG. 30), but can alternatively extendover more or less than half of the beam, and can extend over anysuitable portion of the beam. When the transducing element 120 includesa circular diaphragm, the electrode layers preferably extend over atotal of half the diaphragm area, wherein the electrodes cover thecentral quarter of the diaphragm (e.g. wherein the electrodes arepatterned in a circle, concentric with the diaphragm, that has an areasubstantially equal to 25% of the diaphragm area) and the peripheralquarter of the diaphragm (e.g. wherein the electrodes are patterned inan annulus having an outer diameter substantially equivalent to thediaphragm, wherein the area of the annulus is substantially equal to 25%of the diaphragm area), as shown in FIG. 31. However, the electrodelayers can extend over the diaphragm in any suitable pattern, and coverany suitable proportion of the diaphragm. The electrodes canalternatively be patterned such that they cover less than the entiretyof the transducing element 120, or have any other suitable form factor.The electrodes are preferably aligned along the thickness of thetransducing element 120, but can alternatively be staggered along thethickness of the transducing element 120 or otherwise configured withinthe transducing element.

The casing 200 of the packaged microphone 10 functions to mechanicallyprotect the MEMS microphone chip and to cooperatively define the backcavity 300 with the transducer. The casing 200 can additionally functionto provide an electrical ground. The casing 200 preferably includes aback wall 220, a front wall 240, and side walls 260 extending betweenthe back face and the front face, wherein the transducer is preferablyenclosed within the casing 200, proximal the front wall 240.Alternatively, the microphone can form the front wall 240. A broad faceof the back wall 220 is preferably arranged substantially parallel to abroad face of the front wall 240, but can alternatively be arranged atan angle. The side walls 260 preferably extend perpendicular to the backand/or front walls, but can alternatively extend at any angle. Thecasing 200 can include a rectangular cap with four side walls and a top(e.g., arranged perpendicular to the side walls), include one side wall260, wherein the casing 200 is substantially cylindrical, includemultiple side walls 260, wherein the casing 200 can be polygonal, orhave any other suitable shape. The side walls 260 can be a unitary piecewith the back wall 220, a unitary piece with the front wall 240, be aseparate piece from the front and back walls, be made from a pluralityof separate components, or be otherwise made. The back wall 220 can be aunitary piece with the side wall 260, a cap that couples over or withinthe side walls 260, or any other suitable wall-forming mechanism. Theback wall 220 or front wall 240 can include electrical connectionsconfigured to transmit signals between the packaged microphone 10 andthe external circuitry. The front wall 240 can include an acousticopening 242 through the front wall thickness, wherein the acousticopening 242 can fluidly connect the fluid exterior the casing 200 to thefluid within the front cavity defined between the microphone 100 and thefront wall 240. The microphone 100 is preferably coupled to the frontwall 240 by the substrate, wherein the substrate surface opposing thetransducer is preferably mounted, fixed, adhered, or otherwise fastenedto the front wall 240. However, the microphone 100 can be coupled to thefront wall 240 along a portion of the transducer broad face opposing thesubstrate, such as along a portion of an electrode or along a portion ofthe piezoelectric material. The microphone 100 can be adhered, soldered,welded, or otherwise mounted to the casing 200. The casing 200 ispreferably metal, such as aluminum or steel, but can alternatively beceramic, polymer (e.g., PVC, acrylic, etc.), a combination of the above,or any other suitable material.

The depth of the casing 200 is preferably substantially similar to thedepth required for the back cavity 300, but can alternatively be largeror smaller. The casing 200 preferably defines an internal volume of 50mm³ or less, but can alternatively have a total volume of 50 mm³ orless, define a back volume of 50 mm³ or less, be larger than 50 mm³, orhave any other suitable volume.

The packaged microphone 10 preferably includes a back cavity 300cooperatively defined by the casing 200 and the microphone 100. The backcavity 300 preferably has an acoustic compliance based on the volume ofthe back cavity 300, the density of the fluid filling the back cavity(acoustic medium), and the speed of sound through the fluid filling theback cavity 300. More preferably, the acoustic compliance of the backcavity volume can be calculated as the back cavity volume divided by theproduct of the density of the acoustic medium multiplied by the squareof the speed of sound in the acoustic medium. However, the acousticcompliance of the back cavity can be otherwise calculated. The backcavity compliance can also be adjusted by using methods other thanchanging the volume such as using a flexible wall. The volume of theback cavity 300 is preferably selected for a given application, but canalternatively be selected for a given transducer compliance. The backcavity volume is preferably fixed, but can alternatively be adjustable,wherein the packaged microphone 10 can additionally include a passive oractive adjustment mechanism. The volume of the back cavity preferablyexcludes the volume occupied by the microphone substrate and/or anyassociated electronics (e.g. CMOS, etc.), but can alternatively includethe occupied volume. The acoustic medium is preferably air, but canalternatively be water, gel, or any other suitable acoustic medium.

The following variations of the microphone provide exemplary designsillustrating the above-described optimized microphone, and thediscussion that follows provides additional mathematical and fabricationdetails concerning how the various embodiments can be designed,implemented, and checked for optimization of the ratio noted above.Although optimization of the energy to sensor area ratio and, inparticular, use of the optimization parameter is helpful in determiningpart or all of the sensor geometry, doing so is not necessary, as it issufficient if the resulting microphone, however designed, meets theoptimization criteria described herein.

Single and Stacked Beam Cantilevers

FIG. 3a depicts a beam cantilevered piezoelectric MEMS microphone 30that comprises a multi-layer acoustic sensor having a plurality offingered beams 32, each cantilevered at one of the two left and rightsides 34, 36 of the microphone such the free ends of each facing pair ofbeams 32 are separated by a small gap 38 that can be formed using knownMEMS manufacturing technology. Preferably, this gap is no greater than 3μm, but can be larger depending on the design. For most applications, agap of no greater than 10 μm can be used. A similar gap 40 can be usedbetween adjacent (side-by-side) beams. Cantilevering of the beams 32reduces the influence of material residual stress on the devicebandwidth. Each beam 32 shown in FIG. 3a can be a single, isolated beamthat is interconnected with the other beams to produce the overallmicrophone 30 having the desired capacitance and sensitivitycharacteristics. Alternatively, as shown in FIG. 3b , each beam 32 showncan be the upper beam of a stacked set of two or more beams formed byalternating layers of electrode and piezoelectric material. In FIG. 3b ,there are five alternating layers, although it will be appreciated that,for a stacked beam configuration, additional layers could be used. Thesebeams are constructed without any other layers or materials such thatthe beams comprise only electrode and piezoelectric layers. In theexample shown, the electrode material is molybdenum and thepiezoelectric material is aluminum nitride; however, it will beappreciated that any suitable conductive material can be used for theelectrodes (e.g., titanium) and any suitable piezoelectric material canbe used, such as PZT, ZnO, or others.

The beams 32 can have dimensions determined according to the designmethodology described below to provide a desirable set ofcharacteristics. For some embodiments, the piezoelectric layer can beunder 1 μm, and more preferably, about 0.5 μm, although again this willvary based on a number of factors, including other beam dimensions,materials, etc. For most applications, the beam thickness, and thus thepiezoelectric thickness, will be less than 2 μm, but can go as high as 8μm depending upon the particular application involved. Preferably, thepiezoelectric layer thickness is made as thin as possible whilemaintaining good piezoelectric film quality. For example, the layer canbe made as thin as the available manufacturing technology makes possibleas long as it has sufficient thickness to exhibit a sufficientpiezoelectric effect for the particular application involved. The beamlength should be related to the thickness, as indicated in the designdescription below. The electrode layer can vary as well, but preferablyis on the order of 0.2 μm or less. Preferably, the base end of the beamsare supported with a minimal amount of area to help minimize theresulting capacitance.

The MEMS microphone 30 has several advantageous features, any one ormore of which can be achieved using the design methodologies describedherein. These features include:

1. A maximized or near-maximized ratio of output energy to sensor areafor a given bandwidth, pressure, and piezoelectric material.

2. The ability to design in a desired combination of sensor capacitanceand sensitivity which is achieved by a combination of series andparallel connections between the individual beams. This can be donewithout impacting the overall output energy of the microphone andwithout impacting the input referred piezoelectric noise.

3. The use of adjacent beams separated by a small air gap that providesa high impedance to higher frequency sounds, thereby enabling the deviceto be designed with a lower frequency cutoff. As noted above, this canbe done by keeping the space between adjacent beams (i.e., the gapbetween the facing ends of the beams and/or the gap between the adjacentsides of the beams) to within 10 μm and preferably within 3 μm. Thesegaps can be designed as discussed in Ledermann [15].

4. The use of stacked beams formed only of alternating layers ofelectrode and piezoelectric material.

The design of the cantilever microphone 30 for any particularapplication can be carried out using the design methodology describedbelow. This methodology was developed based on mathematical modeling ofthe beams that was primarily done analytically and verifiedexperimentally. The sensitivity of a single beam has been determined bystarting with equation (20) of Krommer [1] and then determining the beamequation to be:

$\mspace{20mu}{{{\rho\;{A(x)}\frac{\partial^{2}{w\left( {x,t} \right)}}{\partial t^{2\;}}} + {\frac{\partial^{2}}{\partial x^{2}}\left( {\sum\limits_{l = 1}^{N}{M\left( {x,t} \right)}} \right)}} = {f\left( {x,t} \right)}}$  where${\sum\limits_{l = 1}^{N}{M\left( {x,t} \right)}} = {{{- \frac{b}{3}}\frac{\partial^{2}w}{\partial x^{2}}{\sum\limits_{l = 1}^{N}{\frac{1}{s_{11l}}{\left( {1 + \frac{d_{31l}^{2}}{{ɛ_{33l}s_{11l}} - d_{31l}^{2}}} \right)\left\lbrack {\left( {z_{l} - z_{o}} \right)^{3} - \left( {z_{l - 1} - z_{o}} \right)^{3}} \right\rbrack}}}} + {\frac{b}{2}{\sum\limits_{l = 1}^{N}{\frac{1}{s_{11l}}d_{31l}V_{l}\;\frac{\left( {z_{l} - z_{o}} \right)^{2} - \left( {z_{l - 1} - z_{o}} \right)^{2}}{z_{l} - z_{l - 1}}}}}}$

ρ is the density averaged through the thickness, A is thecross-sectional area, w is the beam deflection, t is time, x is thedistance along the beam, M is the beam bending moment, f is the forceper unit width, b is the beam width, N is the number of layers l, s isthe elastic material compliance, d is a piezoelectric couplingcoefficient, ε is the electric permittivity, z is the height from thebottom of the beam as seen in FIG. 3c , and V is the voltage across thelayer. z₀ would be the neutral axis if the beam had no piezoelectricmaterial and can be computed:

$z_{o} = {\frac{1}{2}\frac{\sum\limits_{l = 1}^{N}{\frac{1}{s_{11l}}\left( {z_{l}^{2} - z_{l - 1}^{2}} \right)}}{\sum\limits_{l = 1}^{N}{\frac{1}{s_{11l}}\left( {z_{l} - z_{l - 1}} \right)}}}$

The boundary conditions for the beam equation are:

w = 0@x = 0 $\frac{\partial w}{\partial x} = {{0@x} = 0}$${{\sum\limits_{l = 1}^{N}\;{M\left( {x,t} \right)}} = {{0@x} = L}},{and}$${\frac{\partial}{\partial x}{\sum\limits_{l = 1}^{N}\;{M\left( {x,t} \right)}}} = {{0@x} = {L.}}$

The voltage, V, in the moment equation can be determined by extendingthe method of Irschik [2] to multiple layers resulting in:

$V_{l} = {{- \frac{d_{31l}}{{s_{11l}ɛ_{33l}} - d_{31l}^{2}}}{\quad\left\lbrack {{{\frac{1}{2}\left\lbrack {\left( {z_{l} - z_{o}} \right)^{2} - \left( {z_{l - 1} - z_{o}} \right)^{2}} \right\rbrack}\frac{1}{L}\frac{\partial{w\left( {x = L} \right)}}{\partial x}} + \frac{\sum\limits_{l = 1}^{N}\;{\frac{1}{s_{11l}}d_{31l}{V_{l}\left( {z_{l} - z_{l - 1}} \right)}}}{\sum\limits_{l = 1}^{N}\;{\frac{1}{s_{11l}}\left( {z_{l} - z_{l - 1}} \right)}}} \right\rbrack}}$

The capacitance of a layer is given by:

$C_{l} = \frac{ɛ_{33l}{bL}}{z_{l} - z_{l - 1}}$

The output energy of a layer is calculated by multiplying the square ofthe layer voltage by the layer capacitance:OutputEnergy_(l)=V_(l) ²C_(l)

The device output energy (referred to as the output energy) will be thesum of the output energy of each layer provided that the beams 32 arewired in any combination of series or parallel that preserves thisproduct. In designing and constructing the microphone 30, the parametersof the beam layup (e.g., layer height and length) can be selected suchthat the ratio of this output energy to the sensor area is maximized fora given input pressure, bandwidth, and piezoelectric material. Thisratio is the ratio of:

$\frac{OutputEnergy}{SensorArea}.$

Throughout, the given pressure will be assumed to be unity (1 Pa in SIunits).

Here, sensor area refers to the total chip surface area comprisingpiezoelectric beams. Preferably, the microphone 30 is designed andconstructed to achieve as close to the maximum achievable value aspossible. However, owing to a variety of reasons (e.g., cost ofconstruction), designs of even as low as 10% of the optimal energy tosensor area ratios may be acceptable for certain applications.

It is advantageous to maximize this ratio term for two reasons. First,the output energy remains constant when wiring the beams 32 in series orparallel (allowing the microphone to be matched to a specific circuit).This has been pointed out in the work of Ried [9]. Second, the inputreferred piezoelectric noise remains constant when wiring the beams inseries or parallel. Because both of these characteristics remainconstant, maximizing this ratio can be used as a way to optimize thedesign.

The foregoing equations can be used with beams of arbitrary width andsolved numerically to determine the sensitivity of the beam. For widerbeams (plates), a simple substitution has been suggested by DeVoe [3] toturn the uniaxial stress assumption used above to a plane stressassumption. This substitution is

${\frac{1}{s_{11}} = \frac{1}{{s_{11}\left( {1 - v} \right)}^{2}}},{and}$d₃₁ = d₃₁(1 + v).

However, Elka [4] has shown that the initial uniaxial strain assumptiongives better results when compared to a 3D analytical model or a 3Dfinite element model. If it is assumed that the beam is of constantwidth, the equations simplify significantly and can be solvedanalytically. The assumption of small piezoelectric coupling fromTiersten [5] results in further simplifications. These equations can beused to determine the voltage developed by a specific beam and extendedto determine the voltage developed by several beams, therefore givingthe sensitivity of the piezoelectric microphone. Because beam densityhas been included in these equations, they can also be used to estimatethe bandwidth of the microphone. These equations assume voltage sensingwill be used and that the output of the beams is going into a highimpedance input. Similar equations could be derived if charge sensing isassumed. These equations are also laid out in the works of Krommer [1]and Irschik [2]. For those skilled in the art, these equations could beused in the same manner as those given above to determine an optimizeddevice utilizing charge amplification electronics.

The noise floor (minimum detectable signal) of the piezoelectricmicrophone 30 is limited fundamentally by the dielectric loss angle ofthe material as described by Levinzon [6]. This piezoelectric noise isthermal noise caused by the resistance of the film expressed as:

$\frac{\overset{\_}{v_{n}^{2}}}{\Delta\; f} = {4{kT}\frac{1}{\omega\; C\mspace{14mu}{\tan(\delta)}}}$where v_(n) is the noise spectral density, ∂f is the bandwidth, k isBoltzmann's constant, T is the temperature, ε is the radian frequency, Cis the sensor capacitance, and tan(∂) is the tangent of the dielectricloss angle of the material. This determines the output voltage noise ofa given beam 32 or combination of beams. Other noise sources such asmechanical thermal noise from the beams, radiation impedance of thebeams, and 1/f noise do not dominate the noise of the microphone.

Another significant source of noise is the noise of the accompanyingelectronics. The amplification electronics could be anything rangingfrom a charge amplifier to an integrated circuit for voltageamplification. The demonstrated device uses a junction field effecttransistor (JFET) in a common source amplifier with a load resistor of2.2 kΩ. for amplification because these transistors have relatively lownoise, are small, inexpensive, and relatively simple to model. The JFETnoise can be modeled as shown by Levinzon [7]. At low frequencies, thethermal noise of the resistor R_(b), shown in FIG. 4, dominates thecircuit. A pole is formed at the frequency ω=1/(R_(b)∥R_(p)·C) whereR_(p) is the resistance of the piezoelectric layer obtained from tan(∂).When R_(b) dominates the resistance, a larger capacitance, C, moves thepole to a lower frequency and therefore further attenuates the thermalnoise. A typical noise curve for a piezoelectric sensor connected to aJFET is shown in FIG. 5.

The dynamic range of the microphone 30 exceeds the requirements for mostapplications and will typically be limited by the electronics to whichit is connected. The microphone 30 itself consumes no power so the totalpower consumption is dependent on that of the amplification circuitry.The area of the microphone is determined by the size and number of beamsused and can be traded off with noise floor, sensitivity, and bandwidth.

The sensitivity of this microphone 30 to other parameters such asvibration and temperature has also been investigated. The sensitivity tovibration is related to the material density and thickness as given by:

$\frac{acceleration}{pressure} = \frac{1}{\sum\limits_{l = 1}^{N}\;{\rho_{l}h_{l}}}$

These models were put into Matlab™ and an optimization was performed.The optimization was intended to give a bandwidth in the audible range,a low noise floor, and an area similar to that of commercial MEMSmicrophones.

Because this device 30 uses multiple beams 32, they can be connected ineither series or parallel but the output energy, the product V²C,remains constant for a given acoustic pressure as noted by Ried [9]. Themethod by which these beams are connected illustrates the trade-offbetween sensitivity and noise. If they are all connected in series, thismaximizes sensitivity but the sensor capacitance, C, will be very small.If a JFET is used for amplification, this will increase the frequency ofthe pole filtering the noise and the resulting noise will increase. Ingeneral, a small capacitance will be detrimental because the inputcapacitance to the electronics will act as a capacitive divider andreduce the signal. If all the beams are connected in parallel, thisresults in the minimum sensitivity but maximum sensor capacitance. Anoptimal capacitance, usually between the two limiting cases discussedabove (all parallel v. all series), can be identified to minimize theinput referred noise of the system when using a JFET.

Thus, as will be appreciated by those skilled in the art, area can betraded off with sensitivity and noise floor. More beams consume morearea but result in a larger V²C product. Bandwidth can also be tradedoff with noise floor, sensitivity, and area. Longer beams consume morearea but give a larger V²C product for a given area because they aremore compliant. These longer beams have a lower natural frequency and,therefore, a lower bandwidth.

There are other design/construction factors that influence microphoneoutput. As shown in FIG. 6, beams having a width that is tapered towardtheir free ends can provide a greater V²C output energy. The peak valueof this is at a beam base to tip ratio of about 0.33. Also, for at leastsingle (non-stacked) beams, a layer of Parylene interposed between theelectrode and piezoelectric can provide a better V²C output. Inparticular, after modeling the beam, FIG. 7 was generated to determinethe advantage/disadvantage of an intermediate layer of Parylene. Thisfigure shows that a thin layer of Parylene does slightly enhance the V²Cproduct of a constant area/constant bandwidth group of beams. This thinlayer was not used in the test devices because the Parylene may havehigher surface roughness leading to a reduction in film quality of thetop AlN layer. Because the top layer of AlN would likely be useless, theParylene would need to double the V²C product to be beneficial . . .which it does not. Thus, it may be desirable to limit the use ofParylene to microphone structures using only single (non-stacked) beams.Suitable materials other than Parylene that have a low modulus ofelasticity and low density can be used as well.

After modeling and optimizing the device in Matlab™, devices werefabricated.

Rectangular beams (as opposed to tapered beams) were built for thepurpose of simpler fabrication and testing. The beams were built with a200 nm Mo, 500 nm AlN, 200 nm Mo, 500 nm AlN, 200 nm Mo material stackbecause this combination gives relatively high sensitivity and lownoise.

AlN was selected as the piezoelectric material because it gives equal orsuperior performance compared to other common MEMS piezoelectricmaterials such as ZnO and PZT but is more CMOS compatible than either ofthose two materials. It can be difficult to identify the optimalpiezoelectric material because device performance will depend uponseveral material parameters such as d₃₁, tan(∂), electric permittivity,s, and ρ. These properties depend upon material composition, depositionpower/pressure/temperature, substrate roughness and crystal structure,material thickness, etc. In addition to material deposition variability,it can be difficult to find a source that provides all the necessaryinformation for a complete material comparison as the quoted values forthese parameters can vary substantially, more so for PZT than AlNbecause PZT has more variation in composition and orientation. Using thebest values from the literature [11]-[14] to evaluate both AlN and PZT,they seem to have approximately equal potential for a successful device,although PZT does typically result in a higher sensitivity which couldbe possibly beneficial depending upon the microphone application. AlNparameters in the literature seemed to be more consistent and AlN and Moare also already used in commercial FBAR processes so fabrications withthese materials can more easily be transitioned to a commercial device.Mo was selected because high quality AlN has been deposited on Mo andbecause it worked with the rest of the processing steps. FIG. 8 showshow different electrode materials affect the V²C product. The bestmaterials for this application are those with a low density and lowstiffness. Titanium (Ti), therefore works better than Mo but was notused because of compatibility issues with other processing steps. Thethicknesses of the layers were selected because these were the thinnestthat could be reasonably deposited with good quality. The modelsindicate that thinner layers would be beneficial but were not attemptedin the fabrication.

The processing of the device is shown in FIGS. 9a-9d . First, a 200 nmlayer of SiO₂ was deposited as an etch stop for the DRIE etch. Then a200 nm layer of Mo was deposited, patterned, and etched with dilute AquaRegia (9H₂O:1HNO₃:3HCL). Next, a 500 nm layer of AlN followed by a 200nm layer of Mo was deposited, patterned, and etched with dilute AquaRegia for the Mo and hot (85 C)H₃PO₄ for the AlN. Then another 500 nmAlN and 200 nm Mo were deposited, patterned, and etched. All AlNdepositions were performed at UC Berkeley by Harmonic Devices. DuringAlN deposition, residual stress was monitored in an attempt to limitbeam curvature. Following these etches, both sides of the wafer werecovered with 6 μm of SiO₂ and the back side was patterned and etched forthe DRIE etch to release the beams. Next, the wafer was etched from theback side in an STS DRIE tool. The individual die were then diced with adicing saw and the SiO₂ was removed in 5:1 BHF. Several steps could beimproved upon, most notably, the length of the beams could be more wellcontrolled if an anisotropic silicon etch were used to etch the backcavity and an etch stop was implanted into the silicon under the beams.Some designs utilized an additional metallization step before DRIE toconnect the beams in different combinations of series or parallel andreduce stray capacitance but these devices were not used in this initialproof of concept. A microscope picture of the device can be seen in FIG.10.

After fabricating the devices, they were packaged in a transistoroutline (TO) can as seen in FIG. 11 and wire bonded to a JFET to bufferthe signal. This can be done as indicated in FIG. 4 with the gate inputof the JFET being connected to the sensor electrodes such that signalsreceived from the electrodes are amplified by the transistor. A hole wasdrilled in the TO can below the microphone in order to give opticalaccess to the beams and measure their deflection. This hole also allowsthe size of the back cavity to be adjusted, as the size of the backcavity will determine the low end of the microphone bandwidth. Themicrophone was then placed in a plane wave tube next to a referencemicrophone (Larsen Davis model 2520) and the frequency response wasmeasured using a LabView A/D card and software. This can be seen in FIG.12. The d₃₁ coefficient was measured by actuating the beams andmeasuring the beam curvature with a laser vibrometer. The beamdeflection profile can be seen in FIG. 13.

The natural frequency of the beams was determined by measuring thefrequency response of the beams to actuation. Another parameter thatinfluences the microphone performance is the dielectric loss angletan(∂) of the microphone. This has been measured with both customcircuitry in conjunction with LabView™ software and with an AgilentModel 4284A Precision LCR meter.

For this initial test, only the top layer of AlN was connected to theJFET and only on one side of the beams, thereby resulting in a noisefloor 3 dB higher than would be expected if the entire microphone wereconnected to the JFET. The beams were drawn to be 356 μm but the DRIEetched further than expected, resulting in a natural frequency ofapproximately 11 kHz. This would suggest the beam length is actuallyapproximately 400 μm. The d₃₁ coefficient was measured as 1.68×10⁻¹²N/C. This value is about 65% of the best values quoted in theliterature. The d₃₁ coefficient has been shown to correlate with theX-ray diffraction rocking curve FWHM which, for this layer, is about 2.6degrees while the best reported are around 1 degree. This value islikely higher than others because the layer is only 0.5 μm thick and ontop of other layers. Tan(∂) was measured as 0.04 at 1 kHz. Theliterature typically gives tan(∂) in the range of 0.001 to 0.002 so thisvalue is more than an order of magnitude higher than those typicallyquoted. It was determined that this higher than expected tan(∂) is dueto some residual material left after etching the AlN with H₃PO₄. Aftersome investigation, it was found that the tan(∂) can be reduced bycleaning with Acetone while in an ultrasound bath and heating on a hotplate. The devices with a lower tan(∂) will result in microphones with alower noise floor.

With the measured d₃₁ coefficient and tan(∂) and using the lengthderived from the natural frequency measurement, the microphone modelmatches the measured performance quite well. The sensitivity, shown inFIG. 14, is measured as 0.52 mV/Pa out of the JFET common sourceamplifier with a 2.2 kOhm load resistor. This equates to a raw outputsensitivity of 0.17 mV/Pa for the piezoelectric microphone. The modelpredicts an output sensitivity of 0.18 mV/Pa. The measured inputreferred noise floor for the device is 58.2 dBA while the model predictsan input referred noise floor of 57.3 dBA. FIG. 14 shows the measuredand predicted sensitivities and noise floors. The first peak in themeasured frequency response is caused by the natural frequency of thebeams across from those used in the measurement. They are not exactlythe same length due to a non-uniformity in the DRIE etch.

In the cantilevered beam designs described above, optimization of theoutput energy to sensor area ratio was determined based on a given inputpressure, bandwidth, and piezoelectric material. However, theseconstraints can be taken into account in the design or analysis of apiezoelectric MEMS microphone. In particular, using the optimizationparameter equation:

${{Optimization}\mspace{14mu}{Parameter}} = {\frac{V_{out}^{2}C}{P^{2}A\mspace{14mu}{\tan(\delta)}} \cdot f_{res}^{2}}$the input pressure can be accounted for by the pressure P term, thebandwidth by the f_(res) term, and the characteristics of thepiezoelectric material and the electrode by the dielectric loss angletan(∂). Thus, where a given set of these input constraints is not used,the output energy to sensor area ratio can be optimized by maximizingthe optimization parameter equation given above to take those otherfactors into account.

As one example, consider again a piezoelectric MEMS microphone thatutilizes a rectangular cantilevered beam having one AlN piezoelectriclayer and two Mo electrode layers. For a cantilever, the normalizedoutput energy can be plotted as a function of electrode length as shownin FIG. 15. As the normalized output energy per unit area increases, sowill the optimization parameter so the electrode will go from the baseof the beam to roughly 50% the length of the beam.

When using aluminum nitride as a piezoelectric material, smallpiezoelectric coupling can be assumed. This assumption simplifies theexpression for output voltage from that given above for V₁ to

$V_{out} = {- \frac{{PbL}^{2}d_{31}Z_{Q}}{12{EI}\;\eta\; s_{11}}}$

where P is the pressure amplitude, b is the cantilever width, L is thecantilever length, d₃₁ is the 31 term of the piezoelectric couplingcoefficient matrix, η is the electric permittivity of the piezoelectricmaterial, s₁₁ is the 11 term of the compliance matrix,Z_(Q)=(z_(k)−zn)²−(z_(k−1)−zn)², where zn is the beam neutral axis, thesubscript k refers to the layer and, in this case, refers to thepiezoelectric layer, and EI is the beam bending rigidity given as

${EI} = {\frac{b}{3}{\sum\limits_{k = 1}^{N}\;{\frac{1}{s_{k}}Z_{C_{k}}}}}$where Z_(Ck)=(z_(k)−zn)³−(z_(k−1)−zn)³ and zn is given as

${zn} = {\frac{1}{2}\frac{\sum\limits_{k = 1}^{N}\;{\frac{1}{s_{k}}\left( {z_{k}^{2} - z_{k - 1}^{2}} \right)}}{\sum\limits_{k = 1}^{N}\;\frac{h_{k}}{s_{k}}}}$

The capacitance is approximately

$C = \frac{\eta\; A_{e}}{h_{p}}$where A_(e) is the area covered by the electrode and h_(p) is the heightof the piezoelectric layer. The first resonant frequency isapproximately

$f_{res} = {\frac{1.875^{2}}{2\pi\; L^{2}}\sqrt{\frac{EI}{b{\sum\limits_{k = 1}^{N}\;{\rho_{k}h_{k}}}}}}$

The dielectric loss angle of the microphone is a function of the lossesin the piezoelectric material itself as well as the losses in theelectrodes. This can be approximated as

${\tan(\delta)}_{mic} = {{\tan(\delta)}_{p} + \frac{2\eta_{p}\omega\; L^{2}}{3\sigma_{e}h_{e}h_{p}}}$where the subscripts, p and e, refer to the piezoelectric material andelectrode material respectively, σ is the material conductivity, ω isthe radian frequency, and L is the length of the electrode.

By combining these equations, assuming the length of the electrode isequal to the length of the cantilever beam, the optimization parametercan be calculated as

${OptimizationParameter} = \frac{{bd}_{31p}^{2}Z_{Q_{p}}^{2}\sigma_{e}h_{e}h_{p}}{{3{EIh}_{p}^{2}\eta_{p}s_{11p}^{2}\sigma_{e}h_{e}\mspace{14mu}{\tan(\delta)}_{p}{\sum\limits_{i = 1}^{N}\;{\rho_{i}h_{i}}}} + {{2 \cdot 1.875^{2}}\eta_{p}\sqrt{\frac{EI}{b{\sum\limits_{i = 1}^{N}\;{\rho_{i}h_{i}}}}}}}$

Using this equation and thickness independent material properties, theoptimization would lead to zero thickness layers and an infiniteoptimization parameter. As the molybdenum layer gets thin, however, itsconductivity decreases. Also, very thin AlN tends to have a reducedpiezoelectric coupling coefficient and a large loss angle. For thisreason, these relationships must be included in the optimization.

The d₃₁ data can be extracted by assuming that the d₃₁ coefficientdegrades at the same rate as the d₃₃ coefficient. Plots of d₃₃ andtan(∂) degradation are given in Martin [16] and are shown in FIGS. 16and 17, respectively. Alternatively, the dependence of d₃₁ on thicknesscould be determined experimentally.

The Mo conductivity will also change as the thickness is decreased. Thedependence of Mo thickness with resistivity can be obtained using themodel of Namba [17] to determine this relationship for modelingpurposes. Using this model, a mean free path of 140 nm, P=Q=0, and anRMS surface roughness of 0.5 nm, the relationship between Mo thicknessand resistivity can be determined. The assumed relationships between Moresistivity and Mo thickness, between d₃₁ and AlN thickness, and betweenloss angle and AlN thickness are shown in respective FIGS. 18-20. Usingthe optimization parameter equation and the data from the plots above,the ideal thicknesses for a three layer device are shown in Table 1below.

TABLE 1 Layer Thickness Molybdenum #1   9 nm Aluminum Nitride 1.5 μmMolybdenum #2 1.1 μm

For added accuracy, the fluid loading of air above and below a 1 mm×1 mmdiaphragm has been added to the density summation. The natural frequencyequation can then be used to calculate the length of the beam. For anatural frequency of 20 kHz, the beam will be 374 μm long. The plots ofFIGS. 21-23 show the effect of changing any layer thickness on theoptimization parameter. Small relative changes do not greatly affect thevalue of the optimization parameter except in the case of the bottom Mothickness. For this reason, it may be wise to use a more conservativebottom Mo thickness such as 20 nm. Of course, even more conservativevalues that maintain the optimization parameter above 10% of its maximumobtainable value can be used; thus, electrode thicknesses of 50 nm, 100nm or more can be used since, as shown in FIGS. 18-20 the optimizationparameter, particularly of the bottom electrode, does not decrease toosubstantially with thicknesses in this range. If the desired sensor areais roughly 1 mm×1 mm, this beam can be made to be 1 mm wide and three ofthem can be placed end to end.

This same approach can be used for the stacked beam configuration shownin FIG. 3b of five alternating layers of electrodes and piezoelectricmaterial. Calculated optimal values that maximize the optimizationparameter are given below in Table 2.

TABLE 2 Layer Thickness Molybdenum #1  10 nm Aluminum Nitride #1 1.5 μmMolybdenum #2  10 nm Aluminum Nitride #2 1.5 μm Molybdenum #3  10 nm

The natural frequency equation can then be used to calculate the lengthof the beam. For a natural frequency of 20 kHz, the beam will be 461 μm.The plots of FIGS. 24-26 show the effect of changing any layer thicknesson the optimization parameter. Again, these plots show that theelectrode layer can be increased significantly to, e.g., 20, 50, 100 nmor more without suffering too great a reduction in the calculated outputvoltage to sensor area ratio, and that the middle electrode can bevaried between 5 nm and 1 μm without reducing the ratio to below 10% ofits maximum obtainable value.

Diaphragm Designs

As noted above, rather than a cantilever beam structure, a stressrelieved diaphragm design can also provide a good combination ofsensitivity and low noise floor. Turning now to FIGS. 27a and 27b ,there is shown a piezoelectric MEMS microphone 50 comprising amulti-layer acoustic sensor in the form of a stress-relieved diaphragm52 suspended above a silicon substrate 54. In this embodiment, onlythree layers are used, upper and lower Mo electrode layers, and anintermediate layer of AlN piezoelectric material. However, it will beappreciated that Parylene and other material layers can also be used,and that the diaphragm can have multiple piezoelectric layers such asdiscussed above in connection with the stacked beam cantileverconfigurations. Although the illustrated embodiment includes only threelayers, the upper and lower electrode layers are patterned to eachdefine two independent electrodes. In particular, the first (lower)electrode layer includes a central electrode 56 and an outer ring shapedelectrode 58 that surrounds the central electrode 56. The second (upper)electrode layer shown in FIG. 27a also includes a central electrode 57and an outer ring shaped electrode 59 that surrounds the centralelectrode 57. From the perspective of the top view shown in FIG. 27a ,both the central electrode 57 and outer ring electrode 59 areco-extensive with their associated lower central electrode 56 and ringelectrode 58, respectively. As will be appreciated, the centralelectrodes 56, 57 form a first piezoelectric sensing element and theouter ring electrodes 58, 59 form a second piezoelectric sensingelement. By maintaining the electrodes electrically isolated from eachother, they can be wired together as desired. Since the outer ringpiezoelectric sensing element is strained in the opposite direction asthe central sensing element, the charge produced on these electrodes bythe piezoelectric effect will be of opposite polarity, such that theycan be added together by connecting central electrode 56 to outer ringelectrode 59 and by connecting central electrode 57 to outer ringelectrode 58. The signals from the sensor can be amplified by connectionto a transistor, op-amp or other suitable circuitry in a similar mannerto that discussed above in connection with the cantilever embodiments.

To obtain the stress relieved diaphragm 52, the layers can be formed bydeposition onto a silicon wafer or other suitable substrate 54, with thediaphragm then being micromachined or otherwise processed tosubstantially detach it from the substrate so that the layers can expandor contract as necessary to relieve any residual stress. As shown inFIG. 27a , one way to accomplish this is to use springs 60 to hold thediaphragm 52 in place while it is otherwise released from the substrate54. Once it is stress relieved, the diaphragm 52 can then be reattachedto the substrate 54 about its periphery by any suitable technique, suchas via electrostatic clamping. The springs 60 are created by etchingthrough the AlN to form the border and then undercutting the springs byremoving the material below them. The diaphragm 52 is connected to thesubstrate 54 in the bottom right corner in an area that is used forelectrode leads. The springs in the remaining three corners are thenfixed to the substrate 54 at one end and to the diaphragm 52 at theother. After undercutting the springs, the diaphragm 52 can bereattached to the substrate 54 by holding the bottom, outer electrode 58at ground and applying a voltage bias to the substrate. Thus, thediaphragm 52 has a first portion of its perimeter (at the bottom right)that is attached to the substrate 54 as a direct deposition of at leastone of the layers onto the substrate, and has a second portion of theperimeter attached to the substrate by separate adhesion of the secondportion onto the substrate. It also is connected to the substrate 54 atthe other corners by thin interconnections of one or more of the layersthat act as the springs 60. Electrical connection to the central andouter ring electrodes can be by way of conductive traces 62 that extendacross the piezoelectric layer at the bottom right corner where thediaphragm 52 remains connected to the substrate 54. The optimal layerthicknesses and sizes can be obtained by following the same procedure asabove for the cantilever designs. A reasonable estimate of layerthicknesses can be found by using the same parameters given above;alternatively, a diaphragm model could be used for a more complete andaccurate optimization.

Additional Observations

The fabricated device shows that the models are accurate and only thematerial and processing needs to be improved. When processing anddeposition techniques allow for better material properties to beachieved, the performance will match that shown in FIG. 28. This figureindicates the performance that one could expect for the designed andfabricated devices using a JFET common source amplifier with highquality material parameters. This indicates that this design for apiezoelectric microphone can achieve a noise floor on par with welloptimized capacitive microphones. Some parameters such as sensitivityand power consumption are not included in the FIG. 28 plot because theseparameters are not as significantly interrelated as those given in thefigure. The plus signs in the figure indicate piezoelectric microphonesand the circles indicate capacitive microphones. Piezoelectricmicrophones typically have lower sensitivity than capacitivemicrophones, but this can be corrected by using an application specificintegrated circuit (ASIC) to amplify the signal, as is often used incapacitive microphones. Although this figure assumes high qualitypiezoelectric material, it does not take into account the improvementsthat are possible with the use of a better electrode material, taperedbeams, or a thin compliant layer in the middle of the beam. This alsoassumes a JFET is being used for amplification, thus limiting the noisefloor. An ASIC could have a lower noise floor and improve theperformance of the microphone even further. This also assumes a tan(∂)of 0.001 but it has been shown that tan(∂) can be reduced below thisvalue with proper annealing.

A piezoelectric MEMS microphone constructed as described above couldhave commercial potential competing with electret condenser microphones(ECMs) and MEMS capacitive microphones used in consumer electronics. Thedesign offers performance on par with ECMs and MEMS capacitivemicrophones but offers advantages over each. First, standard ECMs cannotbe mounted to a printed circuit board using the typical lead-free solderprocessing used on all other microchips. This means that they must bespecially attached either by hand or in a more expensive and lessreliable socket. The previously described piezoelectric microphone canwithstand high temperatures and therefore can be mounted using standardtechniques. This piezoelectric microphone is also smaller than ECMs,allowing for a smaller overall electronic device. MEMS capacitivemicrophones also have these advantages and they have, therefore, beenused in cell phones since 2003. MEMS capacitive microphones, however,are more expensive than ECMs due, in large part, to the applicationspecific integrated circuit (ASIC) used to provide readout circuitry tothese microphones. This is a much more expensive part than the JFET usedin ECMs. The piezoelectric MEMS microphone described here can beamplified with a single JFET, therefore, creating a lower costmicrophone with all the advantages of the MEMS capacitive microphone.

Apart from use as an audio microphone, the device can be used for otherapplications such as for ultrasonic detection, with suitable changes inthe design of the microphone structure being used to optimize it forthat application. Also, by covering the beams with an insulatingmaterial such as Parylene (e.g, about 1-2 μm), the microphone can beused as a hydrophone for underwater applications. Similarly, a Paryleneor other suitable insulating covering could be used with the diaphragmdesigns described above to construct a hydrophone, in which case thedevice would include a pressure equalization port or other means ofappropriate pressure equalization with the outside environment, as willbe known by those skilled in the art.

It is to be understood that the foregoing is a description of one ormore preferred exemplary embodiments of the invention. The invention isnot limited to the particular embodiment(s) disclosed herein, but ratheris defined solely by the claims below. Furthermore, the statementscontained in the foregoing description relate to particular embodimentsand are not to be construed as limitations on the scope of the inventionor on the definition of terms used in the claims, except where a term orphrase is expressly defined above. Various other embodiments and variouschanges and modifications to the disclosed embodiment(s) will becomeapparent to those skilled in the art. For example, although sensordesigns that provide at least 10% of the maximum ratio obtainable (ormaximum obtainable optimization parameter) are suitable for manyapplications, more preferable designs will provide at least 25% of themaximum obtainable, and even more preferable designs will provide atleast 50% of the maximum obtainable. In a highly preferred embodiment, adesign using the maximum obtainable optimization parameter can beutilized. All such other embodiments, changes, and modifications areintended to come within the scope of the appended claims.

As used in this specification and claims, the terms “for example,” “forinstance,” “such as,” and “like,” and the verbs “comprising,” “having,”“including,” and their other verb forms, when used in conjunction with alisting of one or more components or other items, are each to beconstrued as open-ended, meaning that the listing is not to beconsidered as excluding other, additional components or items. Otherterms are to be construed using their broadest reasonable meaning unlessthey are used in a context that requires a different interpretation.

REFERENCES

-   [1] Krommer, M., (2001). “On the correction of the Bernoulli-Euler    beam theory for smart piezoelectric beams.” Smart Materials and    Structures, (10), 668-680.-   [2] Irschik, H., Krommer, M., Belyaev, A. K., Schlacher, K., (1998).    “Shaping of Piezoelectric Sensors/Actuators for Vibrations of    Slender Beams: Coupled Theory and Inappropriate Shape Functions.”    Journal of Intelligent Material Systems and Structures, (9),    546-554.-   [3] DeVoe, D. L., Pisano, A. P., (1997). “Modeling and Optimal    Design of Piezoelectric Cantilever Microactuators.” Journal of    Microelectromechanical Systems, (6), 266-270.-   [4] Elka, E., Elata, D., Abramovich, H., (2004). “The    Electromechanical Response of Multilayered Piezoelectric    Structures.” Journal of Microelectromechanical Systems, (13),    332-341.-   [5] Tiersten, H. F. (1969). “Linear Piezoelectric Plate Vibrations.”    (New York: Plenum).-   [6] Levinzon, F. A., (2004). “Fundamental Noise Limit of    Piezoelectric Accelerometer.” IEEE Sensors Journal, (4), 108-111.-   [7] Levinzon, F. A., (2000). “Noise of the JFET Amplifier.” IEEE    Transactions on Circuits and Systems—I: Fundamental Theory and    Applications, (47), 981-985.-   [8] Perkins, N.C. (2001), in: S. G. Braun (Ed.), Nonlinear Systems,    Overview, “Encyclopedia of Vibrations.” Academic Press, 944-951.-   [9] Ried, R. P., Kim, E. S., Hong, D. M., Muller, R. S., (1993).    “Piezoelectric Microphone with On-Chip CMOS Circuits.” Journal of    Microelectromechanical Systems, (2), 111-120.-   [10] Niu, M. N., Kim, E. S., (2003). “Piezoelectric Bimorph    Microphone Built on Micromachined Parylene Diaphragm.” Journal of    Microelectromechanical Systems, (12), 892-898.-   [11] Trolier-McKinstry, S., Muralt, P., (2004). “Thin Film    Piezoelectrics for MEMS.” Journal of Electroceramics, (12), 7-17.-   [12] Tsubouchi, K., Mikoshiba, N., (1985).    “Zero-Temperature-Coefficient SAW Devices on AlN Epitaxial Thin    Films.” IEEE Transactions on Sonics and Ultrasonics, (SU-32),    634-644.-   [13] Dubois, M. A., Muralt, P., (1999). “Properties of aluminum    nitride thin films for piezoelectric transducers and microwave    filter applications.” Applied Physics Letters, (74), 3032-3034.-   [14] Hooker, M. W., (1998). “Properties of PZT-Based Piezoelectric    Ceramics Between −150 and 250° C.” NASA Report: NASA/CR-1998-208708.-   [15] Ledermann, N., Muralt, P., Baborowski, J., Forster, M.,    Pellaux, J.-P., “Piezoelectric Pb(Zrx, Til-x)O3 thin film cantilever    and bridge acoustic sensors for miniaturized photoacoustic gas    detectors.” J. Micromech. Microeng. 14 (2004) 1650-1658.-   [16] Martin, F., Muralt, P., Dubois, M. A., Pezous, A., (2004).    “Thickness dependence of the properties of highly c-axis textured    AlN thin films.” J. Vac. Sci. Technol. A, Vol. 22, No. 2, 361-365.-   [17] Namba, Y., (1970). Japan J. Appl Phys. (9), 1326-1329.

The invention claimed is:
 1. A packaged transducer comprising: amicrophone comprising: a substrate; a transducing element, thetransducing element comprising a first electrode layer, a piezoelectriclayer deposited over the first electrode layer, and a second electrodelayer deposited over piezoelectric material of the piezoelectric layer;and a casing mounted to the microphone and having a front wall and aback wall, wherein a space between the back wall and the transducingelement at least partially defines a back cavity, wherein the substrateis located at least partially between the transducing element and theback wall and the microphone is coupled to the front wall by thesubstrate to at least partially define a front cavity between the frontwall and the microphone, and a fluid connection between the back cavity,the front cavity, and an exterior of the casing.
 2. The transducer ofclaim 1, wherein the microphone comprises a plurality of cantileveredbeams, each beam defining a beam body extending from a beam base andterminating in a beam tip, wherein each beam is joined to the substratealong the respective beam base and is substantially free from thesubstrate along the respective beam body, wherein the plurality of beamsare arranged such that each beam body extends from the respective beambase toward the beam base of an opposing beam, wherein a transduceracoustic compliance of the transducing element is the collectivecompliance of the cantilevered beams.
 3. The transducer of claim 2,wherein the plurality of beams is arranged such that an edge of eachbeam is substantially parallel to an edge of an adjacent beam.
 4. Thetransducer of claim 2, wherein the electrode layers of each beam extendfrom the respective beam base over less than 80% of the respective beambody.
 5. The transducer of claim 2, wherein the beams comprisetriangular beams, wherein the electrode layers extend from the beam baseover less than 80% of the beam body.
 6. The transducer of claim 1,wherein the piezoelectric layer comprises aluminum nitride or doped AlN.7. The transducer of claim 1, further comprising a second piezoelectriclayer deposited directly over the second electrode layer and a thirdelectrode layer deposited directly over the second piezoelectric layer.8. The transducer of claim 1, wherein the transducing element has atransducer acoustic compliance and the back cavity has a back cavityacoustic compliance, and the casing is dimensioned to achieve apredetermined ratio between the transducer and back cavity acousticcompliances.
 9. The transducer of claim 8, wherein the back cavity isdimensioned such that the transducer acoustic compliance is more than17% of the back cavity acoustic compliance.
 10. The transducer of claim8, wherein the back cavity is dimensioned such that the transduceracoustic compliance is less than 581% of the back cavity acousticcompliance.
 11. A packaged transducer comprising: a casing comprising afront wall, a back wall, and a side wall joining the front wall to theback wall; a microphone transducer mounted to the front wall, themicrophone transducer comprising a substrate and a transducing element,the transducing element having a transducer acoustic compliancedependent on the transducing element dimensions; a back cavitycooperatively defined between the back wall, the side wall, the frontwall, and the transducer, the back cavity having a back cavity acousticcompliance; wherein the transducing element comprises a length thatmatches a predetermined resonant frequency and a width, thickness, andelasticity that produces the transducer acoustic compliance within agiven range of the back cavity acoustic compliance.
 12. The transducerof claim 11, wherein the transducer acoustic compliance is between 17%to 538% of the back cavity acoustic compliance.
 13. The transducer ofclaim 12, wherein the transducer acoustic compliance is between 24% to410% of the back cavity acoustic compliance.
 14. The transducer of claim13, wherein the transducer acoustic compliance substantially matches theback cavity acoustic compliance.
 15. The transducer of claim 11, whereinthe transducing element comprises a first electrode layer, anintermediate layer of piezoelectric material deposited over said firstelectrode layer, and a second electrode layer deposited over saidpiezoelectric material.
 16. The transducer of claim 15, wherein thetransducing element comprises a cantilevered beam comprising a beam basejoined to the substrate and a beam body extending from the beam basethat is substantially free from the substrate.
 17. The transducer ofclaim 16, wherein the electrode layers extend from the beam base overless than 80% of the beam body.
 18. The transducer of claim 15, whereinthe transducing element comprises a stress-relieved diaphragm, whereinthe diaphragm is attached to the substrate about a perimeter of thediaphragm at a first and second portion, wherein the first portion ofthe diaphragm comprises a layer of the diaphragm directly deposited ontothe substrate, and the second portion of the diaphragm is adhered to thesubstrate.
 19. The transducer of claim 18, wherein the first and secondelectrode layers define a first and second electrode, respectively,wherein the first and second electrodes each extend over a centralquarter the diaphragm, the first and second electrodes furthercomprising lead traces that extend to the substrate at the first portionof the diaphragm.
 20. The transducer of claim 19, wherein the first andsecond electrode layers further define a third and fourth electrode,respectively, wherein the third and fourth electrodes each extend alongthe entirety of the diaphragm perimeter, have an area substantiallyequivalent to a quarter of the diaphragm area, and are electricallyisolated from the first and second electrodes, respectively.
 21. Thetransducer of claim 20, wherein the first electrode is electricallyconnected to the fourth electrode and the second electrode iselectrically connected to the third electrode.
 22. A packaged transducercomprising: a microphone comprising: a substrate; a transducing element,the transducing element comprising a first electrode layer, apiezoelectric layer deposited over the first electrode layer, and asecond electrode layer deposited over piezoelectric material of thepiezoelectric layer; and a casing mounted to the microphone and having afront wall and a back wall, wherein a space between the back wall andthe transducing element at least partially defines a back cavity,wherein the transducing element has a transducer acoustic compliance andthe back cavity has a back cavity acoustic compliance, and wherein thecasing is dimensioned to achieve a ratio between the transducer and backcavity acoustic compliances.
 23. The transducer of claim 22, wherein thetransducer acoustic compliance is between 17% to 581% of the back cavityacoustic compliance.
 24. The transducer of claim 23, wherein thetransducer acoustic compliance is between 24% to 410% of the back cavityacoustic compliance.